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|Paper IPM / M / 437||
For a finite group G, #Cent(G) denotes the number of centralizers of its elements. A group G is called n-centralizer if #Cent(G)=n, and primitive n-centralizer if #Cent(G) = #Cent([(G)/(Z(G))])=n.
In this paper we compute the number of distinct centralizers of some finite groups and investigate the structure of finite groups with exactly six distinct centralizers. We prove that if G is a 6-centralizer group then [(G)/(Z(G))] ≅ D8,A4,Z2×Z2×Z2 or Z2×Z2×Z2 ×Z2.
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